@misc{10481/109253,
year = {2026},
month = {4},
url = {https://hdl.handle.net/10481/109253},
abstract = {We present an existence and uniqueness result for weak  solutions of Dirichlet boundary value problems governed by  a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to L1(Ω) and to be suitably dominated. We also prove that the solution that we find converges, as s ↗1, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in L1(Ω) and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as s ↗ 1, every classical operator in divergence  form.},
organization = {Ministerio of Ciencia e Innovación (Spain) and European Regional Development Fund (ERDF) - (PID2021-122122NB-I00)},
organization = {Junta de Andalucía - (FQM-116)},
organization = {Juan de la Cierva Fellowship - (JDC2023-050365-I)},
organization = {INdAM-GNAMPA - (grant “Borse di studio per l'estero 2023-2024”)},
organization = {Australian Laureate Fellowship - (FL190100081)},
organization = {Australian Future Fellowship - (FT230100333)},
organization = {Universidad de Granada / CBUA - (Open access charge)},
publisher = {Elsevier},
keywords = {Nonlocal operator},
keywords = {Divergence form operator},
keywords = {Dirichlet boundary condition},
title = {Nonlocal operators in divergence form and existence theory for integrable data},
doi = {10.1016/j.jfa.2025.111317},
author = {Arcoya Álvarez, David and Dipierro, Serena and Proietti Lippi, Edoardo and Sportelli, Caterina and Valdinoci, Enrico},
}